Asymptotic expansion of perturbative Chern-Simons theory via Wiener space
After Witten's seminal work on relations between quantum link invariants and heuristic Chern-Simons path integrals there has been a sustained effort to make rigorous mathematical sense of the latter. In the context of white noise analysis the non-existence of an infinite-dimensional Lebesgue measure on the space of connections is circumvented by using a suitable Gaussian measure instead. Integration is then replaced by pairing functionals of distributional connections with regularized loop holonomies. An additional difficulty in the Chern-Simons case is that the Gaussian kernel is degenerate and a gauge is usually fixed before making sense of the integrals. Unfortunately, this defeats the original purpose of Witten's work, which was to find an intrinsic 3D representation for quantum link invariants. However, no alternative approach has emerged so far in the non-Abelian case. This paper is one of the first attempts to define integrals appearing in the formal expansion of Chern-Simons integrals in powers of the level \(k\) along similar lines. Heuristically, these integrals correspond to Vassiliev (finite-type) invariants rather than link polynomials. Unitary bundles of connections over general closed 3-manifolds are considered, and the expansion is centered at a flat reference connection assumed isolated and irreducible. The Lorentz gauge is chosen, and the usual trick with anti-commuting variables is used to integrate out the degeneracy and obtain a formal expression for the Faddeev-Popov determinant. The reduced kernel then becomes elliptic and a regularized determinant can be defined. Since the Gaussian measure is not supported by smooth connections the usual loop holonomies (Wilson loops) are not integrable with respect to it. Hence regularizations are defined by smearing Wilson loops into smooth currents via convolution. Under additional technical assumtions on the reference connection the authors show that partial sums of the regularized series make sense as Gaussian integrals. The main result gives the answer as an asymptotic expansion in powers of \(k\) along with an explicit estimate on the tail for \(k\to\infty\). Dependence of the answer on regularization (the framing anomaly) is not addressed, and explicit calculation of the integrals is deferred to future work. The paper is very technical with many pages covered almost exclusively by formulas.
scientific article; zbMATH DE number 5552986
scientific article